Integrand size = 85, antiderivative size = 15 \[ \int \frac {e^x (-1-x)-3 x^2+\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )}{\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )} \, dx=x+\frac {1}{\log \left (\log \left (e^x x+x^3\right )\right )} \] Output:
1/ln(ln(exp(x)*x+x^3))+x
Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {e^x (-1-x)-3 x^2+\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )}{\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )} \, dx=x+\frac {1}{\log \left (\log \left (x \left (e^x+x^2\right )\right )\right )} \] Input:
Integrate[(E^x*(-1 - x) - 3*x^2 + (E^x*x + x^3)*Log[E^x*x + x^3]*Log[Log[E ^x*x + x^3]]^2)/((E^x*x + x^3)*Log[E^x*x + x^3]*Log[Log[E^x*x + x^3]]^2),x ]
Output:
x + Log[Log[x*(E^x + x^2)]]^(-1)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (x^3+e^x x\right ) \log \left (x^3+e^x x\right ) \log ^2\left (\log \left (x^3+e^x x\right )\right )-3 x^2+e^x (-x-1)}{\left (x^3+e^x x\right ) \log \left (x^3+e^x x\right ) \log ^2\left (\log \left (x^3+e^x x\right )\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {(x-2) x}{\left (x^2+e^x\right ) \log \left (x^3+e^x x\right ) \log ^2\left (\log \left (x^3+e^x x\right )\right )}+\frac {x \log \left (x \left (x^2+e^x\right )\right ) \log ^2\left (\log \left (x \left (x^2+e^x\right )\right )\right )-x-1}{x \log \left (x^3+e^x x\right ) \log ^2\left (\log \left (x^3+e^x x\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {1}{\log \left (x^3+e^x x\right ) \log ^2\left (\log \left (x^3+e^x x\right )\right )}dx-\int \frac {1}{x \log \left (x^3+e^x x\right ) \log ^2\left (\log \left (x^3+e^x x\right )\right )}dx+2 \int \frac {x}{\left (-x^2-e^x\right ) \log \left (x^3+e^x x\right ) \log ^2\left (\log \left (x^3+e^x x\right )\right )}dx+\int \frac {x^2}{\left (x^2+e^x\right ) \log \left (x^3+e^x x\right ) \log ^2\left (\log \left (x^3+e^x x\right )\right )}dx+x\) |
Input:
Int[(E^x*(-1 - x) - 3*x^2 + (E^x*x + x^3)*Log[E^x*x + x^3]*Log[Log[E^x*x + x^3]]^2)/((E^x*x + x^3)*Log[E^x*x + x^3]*Log[Log[E^x*x + x^3]]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(29\) vs. \(2(14)=28\).
Time = 3.39 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.00
method | result | size |
parallelrisch | \(-\frac {-2-2 x \ln \left (\ln \left (x \left (x^{2}+{\mathrm e}^{x}\right )\right )\right )}{2 \ln \left (\ln \left (x \left (x^{2}+{\mathrm e}^{x}\right )\right )\right )}\) | \(30\) |
risch | \(x +\frac {1}{\ln \left (\ln \left (x \right )+\ln \left (x^{2}+{\mathrm e}^{x}\right )-\frac {i \pi \,\operatorname {csgn}\left (i x \left (x^{2}+{\mathrm e}^{x}\right )\right ) \left (-\operatorname {csgn}\left (i x \left (x^{2}+{\mathrm e}^{x}\right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (x^{2}+{\mathrm e}^{x}\right )\right )+\operatorname {csgn}\left (i \left (x^{2}+{\mathrm e}^{x}\right )\right )\right )}{2}\right )}\) | \(74\) |
Input:
int(((exp(x)*x+x^3)*ln(exp(x)*x+x^3)*ln(ln(exp(x)*x+x^3))^2+(-1-x)*exp(x)- 3*x^2)/(exp(x)*x+x^3)/ln(exp(x)*x+x^3)/ln(ln(exp(x)*x+x^3))^2,x,method=_RE TURNVERBOSE)
Output:
-1/2*(-2-2*x*ln(ln(x*(x^2+exp(x)))))/ln(ln(x*(x^2+exp(x))))
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.80 \[ \int \frac {e^x (-1-x)-3 x^2+\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )}{\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )} \, dx=\frac {x \log \left (\log \left (x^{3} + x e^{x}\right )\right ) + 1}{\log \left (\log \left (x^{3} + x e^{x}\right )\right )} \] Input:
integrate(((exp(x)*x+x^3)*log(exp(x)*x+x^3)*log(log(exp(x)*x+x^3))^2+(-1-x )*exp(x)-3*x^2)/(exp(x)*x+x^3)/log(exp(x)*x+x^3)/log(log(exp(x)*x+x^3))^2, x, algorithm="fricas")
Output:
(x*log(log(x^3 + x*e^x)) + 1)/log(log(x^3 + x*e^x))
Time = 0.57 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {e^x (-1-x)-3 x^2+\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )}{\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )} \, dx=x + \frac {1}{\log {\left (\log {\left (x^{3} + x e^{x} \right )} \right )}} \] Input:
integrate(((exp(x)*x+x**3)*ln(exp(x)*x+x**3)*ln(ln(exp(x)*x+x**3))**2+(-1- x)*exp(x)-3*x**2)/(exp(x)*x+x**3)/ln(exp(x)*x+x**3)/ln(ln(exp(x)*x+x**3))* *2,x)
Output:
x + 1/log(log(x**3 + x*exp(x)))
Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (14) = 28\).
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.93 \[ \int \frac {e^x (-1-x)-3 x^2+\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )}{\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )} \, dx=\frac {x \log \left (\log \left (x^{2} + e^{x}\right ) + \log \left (x\right )\right ) + 1}{\log \left (\log \left (x^{2} + e^{x}\right ) + \log \left (x\right )\right )} \] Input:
integrate(((exp(x)*x+x^3)*log(exp(x)*x+x^3)*log(log(exp(x)*x+x^3))^2+(-1-x )*exp(x)-3*x^2)/(exp(x)*x+x^3)/log(exp(x)*x+x^3)/log(log(exp(x)*x+x^3))^2, x, algorithm="maxima")
Output:
(x*log(log(x^2 + e^x) + log(x)) + 1)/log(log(x^2 + e^x) + log(x))
Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.80 \[ \int \frac {e^x (-1-x)-3 x^2+\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )}{\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )} \, dx=\frac {x \log \left (\log \left (x^{3} + x e^{x}\right )\right ) + 1}{\log \left (\log \left (x^{3} + x e^{x}\right )\right )} \] Input:
integrate(((exp(x)*x+x^3)*log(exp(x)*x+x^3)*log(log(exp(x)*x+x^3))^2+(-1-x )*exp(x)-3*x^2)/(exp(x)*x+x^3)/log(exp(x)*x+x^3)/log(log(exp(x)*x+x^3))^2, x, algorithm="giac")
Output:
(x*log(log(x^3 + x*e^x)) + 1)/log(log(x^3 + x*e^x))
Time = 2.49 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {e^x (-1-x)-3 x^2+\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )}{\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )} \, dx=x+\frac {1}{\ln \left (\ln \left (x\,{\mathrm {e}}^x+x^3\right )\right )} \] Input:
int(-(exp(x)*(x + 1) + 3*x^2 - log(x*exp(x) + x^3)*log(log(x*exp(x) + x^3) )^2*(x*exp(x) + x^3))/(log(x*exp(x) + x^3)*log(log(x*exp(x) + x^3))^2*(x*e xp(x) + x^3)),x)
Output:
x + 1/log(log(x*exp(x) + x^3))
\[ \int \frac {e^x (-1-x)-3 x^2+\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )}{\left (e^x x+x^3\right ) \log \left (e^x x+x^3\right ) \log ^2\left (\log \left (e^x x+x^3\right )\right )} \, dx=-\left (\int \frac {e^{x}}{e^{x} {\mathrm {log}\left (\mathrm {log}\left (e^{x} x +x^{3}\right )\right )}^{2} \mathrm {log}\left (e^{x} x +x^{3}\right ) x +{\mathrm {log}\left (\mathrm {log}\left (e^{x} x +x^{3}\right )\right )}^{2} \mathrm {log}\left (e^{x} x +x^{3}\right ) x^{3}}d x \right )-\left (\int \frac {e^{x}}{e^{x} {\mathrm {log}\left (\mathrm {log}\left (e^{x} x +x^{3}\right )\right )}^{2} \mathrm {log}\left (e^{x} x +x^{3}\right )+{\mathrm {log}\left (\mathrm {log}\left (e^{x} x +x^{3}\right )\right )}^{2} \mathrm {log}\left (e^{x} x +x^{3}\right ) x^{2}}d x \right )-3 \left (\int \frac {x}{e^{x} {\mathrm {log}\left (\mathrm {log}\left (e^{x} x +x^{3}\right )\right )}^{2} \mathrm {log}\left (e^{x} x +x^{3}\right )+{\mathrm {log}\left (\mathrm {log}\left (e^{x} x +x^{3}\right )\right )}^{2} \mathrm {log}\left (e^{x} x +x^{3}\right ) x^{2}}d x \right )+x \] Input:
int(((exp(x)*x+x^3)*log(exp(x)*x+x^3)*log(log(exp(x)*x+x^3))^2+(-1-x)*exp( x)-3*x^2)/(exp(x)*x+x^3)/log(exp(x)*x+x^3)/log(log(exp(x)*x+x^3))^2,x)
Output:
- int(e**x/(e**x*log(log(e**x*x + x**3))**2*log(e**x*x + x**3)*x + log(lo g(e**x*x + x**3))**2*log(e**x*x + x**3)*x**3),x) - int(e**x/(e**x*log(log( e**x*x + x**3))**2*log(e**x*x + x**3) + log(log(e**x*x + x**3))**2*log(e** x*x + x**3)*x**2),x) - 3*int(x/(e**x*log(log(e**x*x + x**3))**2*log(e**x*x + x**3) + log(log(e**x*x + x**3))**2*log(e**x*x + x**3)*x**2),x) + x